Casimir Forces Between Bodies
The Casimir force on an arbitrary arrangement of magnetoelectric bodies is calculated in two different ways. A direct evaluation of the ground-state average Lorentz force on the internal charge and current densities of a body gives the Casimir force as a
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Casimir Forces Between Bodies
We begin our studies of dispersion forces in this chapter by considering Casimir forces between bodies. Casimir forces can be obtained from an entirely macroscopic calculation, so their treatment is conceptually simpler than that of dispersion forces on atoms, where atom–field couplings need to be taken into account. Technically, they are quite involved due to an extensive use of tensor calculus. We will first derive two equivalent general formulae for the Casimir force on a magnetoelectric body of arbitrary shape and material due to the presence of other such bodies: starting from the Lorentz force, the Casimir force follows as a volume integral over force densities; while an equivalent formulation via the Maxwell stress tensor leads to the Casimir force as a surface integral over stresses. We use the general results to obtain the force between two plates, thus recovering the famous results of Casimir and Lifshitz. To begin with, let us briefly recall the common physical origin of all dispersion forces. As pointed out in Chap. 1, they are effective electromagnetic forces due to quantum ground-state fluctuations. The total Lorentz force on an arbitrary body or atom, characterised by a charge density ρˆ and a current density ˆj and occupying a volume V , is given by ˆ = F
ˆ + ˆj × B ˆ d3r ρˆ E
(3.1)
V
where the electric field acts on the charge density and the magnetic field acts on the current density. For an electrically neutral, unpolarised system in its ground state with no net charges or currents, one has ρ ˆ = 0 and ˆj = 0 for the quantum averages. Likewise, when the quantum electromagnetic field is in its vacuum state and no external fields are present, the quantum averages of electric and magnetic fields also ˆ = B ˆ = 0, see Fig. 3.1(i). In the absence of correlations, this would vanish, E immediately imply that the Lorentz force on the body also vanishes on the quantum average. However, both the body and the electromagnetic fields are subject to nonˆ 2 , B ˆ 2 = 0. These fluctuations vanishing ground-state fluctuations, ρˆ2 , ˆj 2 , E S. Y. Buhmann, Dispersion Forces I, Springer Tracts in Modern Physics 247, DOI: 10.1007/978-3-642-32484-0_3, © Springer-Verlag Berlin Heidelberg 2012
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3 Casimir Forces Between Bodies
(i)
(ii)
(iii)
(iv)
(v)
Fig. 3.1 Correlated fluctuations. (i) A neutral unpolarised body in the absence of electromagnetic fields. (ii) Fluctuations of the charge density induce (iii) correlated electric fields. (iv) Fluctuating electric fields induce (v) a correlated charge density
are mutually correlated: for instance, a fluctuating charge density (Fig. 3.1(ii)) acts as a source for electric fields. The latter are not independent, but precisely determined by their fluctuating sources (Fig. 3.1(iii)). These correlated electric fields then act back on the charge density to give rise to a non-vanishing net force. The inverse process is also possible, where fluctuating electric fields (Fig. 3.1(iv)) induce correlat
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